Notwithstanding, both the relativistic and non-relativistic torsional anomalies can appear with the (usually) infrared (IR) temperature scale or chemical potentials, similarly to the gravitational contribution to the CVE and we focus here on these terms. Moreover, such a finite temperature (chemical potential) term can be “universal" as it rest on the anomaly related quantum statistical properties, with the same caveats as the hydrodynamic CVE (CME). In this paper we discuss the chiral torsional effect (CTE) from the perspective of the analogy and similarity with CVE in relativistic systems with chiral fermions. We find that the coefficient is non-zero and directly related to that of the CVE (at the level of linear response), and, in accordance with the previous results found with e.g. the simple Landau level approach to chiral torsional anomalies. In particular, recent papers [1]}, [2]} finding no torsional contribution can be easily incorporated with our framework by noting that when computing the linear response around flat space, torsion has been actually set to zero with only non-zero metric variations. This is equivalent to whether or not frame fields and torsion appear in the hydrodynamics and needs to be separately addressed at the level of the constitutive relations in the presence of sources. Nevertheless, we find non-zero transport coefficients in the case torsionful sources do appear.
